Mister Exam
-2cosx/3< inequation
The solution
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-2*cos(x)
--------- < 0
3
$$\frac{\left(-1\right) 2 \cos{\left(x \right)}}{3} < 0$$
(-2*cos(x))/3 < 0
Detail solution
Given the inequality:
$$\frac{\left(-1\right) 2 \cos{\left(x \right)}}{3} < 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{\left(-1\right) 2 \cos{\left(x \right)}}{3} = 0$$
Solve:
Given the equation
$$\frac{\left(-1\right) 2 \cos{\left(x \right)}}{3} = 0$$
- this is the simplest trigonometric equation
We get:
$$\frac{\left(-1\right) 2 \cos{\left(x \right)}}{3} = 0$$
The equation is transformed to
$$\cos{\left(x \right)} = 0$$
This equation is transformed to
$$x = \pi n + \operatorname{acos}{\left(0 \right)}$$
$$x = \pi n - \pi + \operatorname{acos}{\left(0 \right)}$$
Or
$$x = \pi n + \frac{\pi}{2}$$
$$x = \pi n - \frac{\pi}{2}$$
, where n - is a integer
$$x_{1} = \pi n + \frac{\pi}{2}$$
$$x_{2} = \pi n - \frac{\pi}{2}$$
$$x_{1} = \pi n + \frac{\pi}{2}$$
$$x_{2} = \pi n - \frac{\pi}{2}$$
This roots
$$x_{1} = \pi n + \frac{\pi}{2}$$
$$x_{2} = \pi n - \frac{\pi}{2}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(\pi n + \frac{\pi}{2}\right) + - \frac{1}{10}$$
=
$$\pi n - \frac{1}{10} + \frac{\pi}{2}$$
substitute to the expression
$$\frac{\left(-1\right) 2 \cos{\left(x \right)}}{3} < 0$$
$$\frac{\left(-1\right) 2 \cos{\left(\pi n - \frac{1}{10} + \frac{\pi}{2} \right)}}{3} < 0$$
but
Then
$$x < \pi n + \frac{\pi}{2}$$
no execute
one of the solutions of our inequality is:
$$x > \pi n + \frac{\pi}{2} \wedge x < \pi n - \frac{\pi}{2}$$
$$\frac{\left(-1\right) 2 \cos{\left(x \right)}}{3} < 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{\left(-1\right) 2 \cos{\left(x \right)}}{3} = 0$$
Solve:
Given the equation
$$\frac{\left(-1\right) 2 \cos{\left(x \right)}}{3} = 0$$
- this is the simplest trigonometric equation
with the change of sign in 0
We get:
$$\frac{\left(-1\right) 2 \cos{\left(x \right)}}{3} = 0$$
Divide both parts of the equation by -2/3
The equation is transformed to
$$\cos{\left(x \right)} = 0$$
This equation is transformed to
$$x = \pi n + \operatorname{acos}{\left(0 \right)}$$
$$x = \pi n - \pi + \operatorname{acos}{\left(0 \right)}$$
Or
$$x = \pi n + \frac{\pi}{2}$$
$$x = \pi n - \frac{\pi}{2}$$
, where n - is a integer
$$x_{1} = \pi n + \frac{\pi}{2}$$
$$x_{2} = \pi n - \frac{\pi}{2}$$
$$x_{1} = \pi n + \frac{\pi}{2}$$
$$x_{2} = \pi n - \frac{\pi}{2}$$
This roots
$$x_{1} = \pi n + \frac{\pi}{2}$$
$$x_{2} = \pi n - \frac{\pi}{2}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(\pi n + \frac{\pi}{2}\right) + - \frac{1}{10}$$
=
$$\pi n - \frac{1}{10} + \frac{\pi}{2}$$
substitute to the expression
$$\frac{\left(-1\right) 2 \cos{\left(x \right)}}{3} < 0$$
$$\frac{\left(-1\right) 2 \cos{\left(\pi n - \frac{1}{10} + \frac{\pi}{2} \right)}}{3} < 0$$
2*sin(-1/10 + pi*n)
------------------- < 0
3 but
2*sin(-1/10 + pi*n)
------------------- > 0
3 Then
$$x < \pi n + \frac{\pi}{2}$$
no execute
one of the solutions of our inequality is:
$$x > \pi n + \frac{\pi}{2} \wedge x < \pi n - \frac{\pi}{2}$$
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x1 x2
Solving inequality on a graph
Rapid solution
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/ / pi\ / 3*pi \\ Or|And|0 <= x, x < --|, And|x <= 2*pi, ---- < x|| \ \ 2 / \ 2 //
$$\left(0 \leq x \wedge x < \frac{\pi}{2}\right) \vee \left(x \leq 2 \pi \wedge \frac{3 \pi}{2} < x\right)$$
((0 <= x)∧(x < pi/2))∨((x <= 2*pi)∧(3*pi/2 < x))
Rapid solution 2
[src]
pi 3*pi
[0, --) U (----, 2*pi]
2 2
$$x\ in\ \left[0, \frac{\pi}{2}\right) \cup \left(\frac{3 \pi}{2}, 2 \pi\right]$$
x in Union(Interval.Ropen(0, pi/2), Interval.Lopen(3*pi/2, 2*pi))